Approximate perfect differential equations of second order

Abstract

In this paper we prove the Hyers-Ulam stability of the perfect linear differential equation $f(t)y^{″}(t)+f_1(t)y^{\prime }(t)+f_2(t)y(t)=Q(t)$, where $f,y\in C^2[a,b]$, $Q\in C[a,b]$, $f_2(t)=f_1^{\prime }(t)-f^{″}(t)$ and $-\mathrm{\infty }<a<b<+\mathrm{\infty }$.

MSC:34K20, 26D10, 39B82, 34K06, 39B72.

1 Introduction

The question concerning the stability of group homomorphisms was posed by Ulam [1]. Hyers [2] solved the case of approximately additive mappings in Banach spaces and T.M. Rassias generalized the result of Hyers [3].

Definition 1.1 Let X be a normed space over a scalar field and let I be an open interval. Assume that $a_0,a_1,\dots ,a_n$, $h:I\to \mathbb{K}$ are continuous functions. We say that the differential equation

$$a_n(t)y^{(n)}(t)+a_{n-1}(t)y^{(n-1)}(t)+\cdots +a_1(t)y^{\prime }(t)+a_0(t)y(t)+h(t)=0$$

(1.1)

has the Hyers-Ulam stability if, for any function $f:I\to X$ satisfying the differential inequality

$$\parallel a_n(t)y^{(n)}(t)+a_{n-1}(t)y^{(n-1)}(t)+\cdots +a_1(t)y^{\prime }(t)+a_{0}y(t)+h(t)\parallel \le \epsilon$$

for all $t\in I$ and some $\epsilon \ge 0$, there exists a solution $g:I\to X$ of (1.1) such that $\parallel f(t)-g(t)\parallel \le K(\epsilon )$ for all $t\in I$, where $K(\epsilon )$ is a function depending only on ε.

Obłoza [4, 5] was the first author who investigated the Hyers-Ulam stability of differential equations (also see [6]).

Jung [7] solved the inhomogeneous differential equation of the form $y^{″}+2x{y}^{\prime }-2ny={\sum }_{m=0}^{\mathrm{\infty }}{a}_m{x}^m$, where n is a positive integer, and he used this result to prove the Hyers-Ulam stability of the differential equation $y^{″}+2x{y}^{\prime }-2ny=0$ in a special class of analytic functions.

Li and Shen [8] proved that if the characteristic equation ${\lambda }^2+\alpha \lambda +\beta =0$ has two different positive roots, then the linear differential equation of second order with constant coefficients $y^{″}(x)+\alpha y^{\prime }(x)+\beta y(x)=f(x)$ has the Hyers-Ulam stability where $y\in C^2[a,b]$, $f\in C[a,b]$ and $-\mathrm{\infty }<a<b<+\mathrm{\infty }$ (see also [9, 10]). Abdollahpour and Najati [11] proved that the third-order differential equation $y^{(3)}(t)+\alpha y^{″}(t)+\beta y^{\prime }(t)+\gamma y(t)=f(t)$ has the Hyers-Ulam stability. Ghaemi et al. [12] proved the Hyers-Ulam stability of the exact second-order linear differential equation

$$p_0(x){\gamma }^{″}+p_1(x){\gamma }^{\prime }+p_2(x)\gamma +f(x)=0$$

with $p_0^{″}(x)-p_1^{\prime }(x)+p_2(x)=0$. Here $p_0$, $p_1$, $p_2$, $f:(a,b)\to \mathbb{R}$ are continuous functions. For more results about the Hyers-Ulam stability of differential equations, we can refer to [13–21].

Definition 1.2 We say that the differential equation

$$f(t)y^{″}(t)+f_1(t)y^{\prime }(t)+f_2(t)y(t)=Q(t),$$

(1.2)

is perfect if it can be written as $\frac{d}{dt}[f(t)y^{\prime }(t)+(f_1(t)-f^{\prime }(t))y(t)]=Q(t)$.

It is clear that the differential equation (1.2) is perfect if and only if $f_2(t)=f_1^{\prime }(t)-f^{″}(t)$. The aim of this paper is to investigate the Hyers-Ulam stability of the perfect differential equation (1.2), where $f,y\in C^2[a,b]$, $Q\in C[a,b]$, $f_1\in C^1[a,b]$, $f_2(t)=f_1^{\prime }(t)-f^{″}(t)$ and $-\mathrm{\infty }<a<b<+\mathrm{\infty }$. More precisely, we prove that the equation (1.2) has the Hyers-Ulam stability.

2 Hyers-Ulam stability of the perfect differential equation $f(t)y^{″}(t)+f_1(t)y^{\prime }(t)+f_2(t)y(t)=Q(t)$

In the following theorem, we prove the Hyers-Ulam stability of the differential equation (1.2).

Throughout this section, a and b are real numbers with $-\mathrm{\infty }<a<b<+\mathrm{\infty }$.

Theorem 2.1 The perfect differential equation

$$f(t)y^{″}(t)+f_1(t)y^{\prime }(t)+f_2(t)y(t)=Q(t)$$

has the Hyers-Ulam stability, where $f,y\in C^2[a,b]$, $f_1\in C^1[a,b]$, $Q\in C[a,b]$ and $f(t)\ne 0$ for all $t\in [a,b]$.

Proof Let $\epsilon >0$ and $y\in C^2[a,b]$ with

$$|f(t)y^{″}(t)+f_1(t)y^{\prime }(t)+f_2(t)y(t)-Q(t)|⩽\epsilon .$$

Let $g(t)=f(t)y^{\prime }+(f_1(t)-f^{\prime }(t))y$ for all $t\in [a,b]$. It is clear that

$$|g^{\prime }(t)-Q(t)|=|f(t)y^{″}(t)+f_1(t)y^{\prime }(t)+f_2(t)y(t)-Q(t)|⩽\epsilon .$$

We define

$$z(x)=g(b)-{\int }_x^{b}Q(t)dt,x\in [a,b].$$

Then

$$z^{\prime }(x)=Q(x),x\in [a,b].$$

(2.1)

Also, we have

$$\begin{array}{rl}|z(x)-g(x)|& =|g(b)-g(x)-{\int }_x^{b}Q(t)dt|=|{\int }_x^b{g}^{\prime }(t)dt-{\int }_x^{b}Q(t)dt|\\ ⩽{\int }_x^b|g^{\prime }(t)-Q(t)|dt⩽\epsilon (b-a)\end{array}$$

for all $x\in [a,b]$. Now we define

$$F(x)=\frac{1}{f(x)}exp\left\{{\int }_a^x\frac{f_1(t)}{f(t)}dt\right\},u(x)=\frac{y(b)F(b)}{F(x)}-\frac{1}{F(x)}{\int }_x^b\frac{z(t)F(t)}{f(t)}dt$$

for all $x\in [a,b]$. It is clear that $u\in C^2[a,b]$ and

$$u^{\prime }(x)F(x)+u(x)F^{\prime }(x)=\frac{z(x)F(x)}{f(x)},F^{\prime }(x)=\frac{f_1(x)-f^{\prime }(x)}{f(x)}F(x).$$

Therefore,

$$f(x)u^{\prime }(x)+[f_1(x)-f^{\prime }(x)]u(x)=z(x),x\in [a,b].$$

(2.2)

Hence, (2.1) implies that

$$f(x)u^{″}(x)+f_1(x)u^{\prime }(x)+f_2(x)u(x)=Q(x),x\in [a,b].$$

Also, we have

$$\begin{array}{rl}|y(x)-u(x)|& =|y(x)-\frac{y(b)F(b)}{F(x)}+\frac{1}{F(x)}{\int }_x^b\frac{z(t)F(t)}{f(t)}dt|\\ =\frac{1}{|F(x)|}|y(x)F(x)-y(b)F(b)+{\int }_x^b\frac{z(t)F(t)}{f(t)}dt|\\ =\frac{1}{|F(x)|}|{\int }_x^b\frac{z(t)F(t)}{f(t)}dt-{\int }_x^b{[y(t)F(t)]}^{\prime }dt|\\ =\frac{1}{|F(x)|}\left|{\int }_x^b(\frac{z(t)F(t)}{f(t)}-y^{\prime }(t)F(t)-y(t)F^{\prime }(t))dt\right|\\ =\frac{1}{|F(x)|}|{\int }_x^{b}F(t)(\frac{z(t)}{f(t)}-y^{\prime }(t)-\frac{f_1(t)-f^{\prime }(t)}{f(t)}y(t))dt|\\ ⩽\frac{1}{|F(x)|}{\int }_x^b\left|\frac{F(t)}{f(t)}\right||z(t)-y^{\prime }(t)f(t)-[f_1(t)-f^{\prime }(t)]y(t)|dt\\ =\frac{1}{|F(x)|}{\int }_x^b\left|\frac{F(t)}{f(t)}\right||z(t)-g(t)|dt\\ ⩽\epsilon (b-a)\frac{1}{|F(x)|}{\int }_x^b\left|\frac{F(t)}{f(t)}\right|dt\end{array}$$

(2.3)

for all $x\in [a,b]$. Since $\frac{f_1}{f}\in C[a,b]$, there exist constants $m^{\prime }$ and $M^{\prime }$ such that $m^{\prime }⩽\frac{f_1(x)}{f(x)}⩽M^{\prime }$. Thus

$$\{\begin{array}{ll}1⩽exp\{{\int }_a^x\frac{f_1(t)}{f(t)}dt\}⩽e^{M^{\prime }(b-a)}& \text{if }{m}^{\prime }\ge 0;\\ e^{m^{\prime }(b-a)}⩽exp\{{\int }_a^x\frac{f_1(t)}{f(t)}dt\}⩽e^{M^{\prime }(b-a)}& \text{if }{m}^{\prime }<0⩽M^{\prime };\\ e^{m^{\prime }(b-a)}⩽exp\{{\int }_a^x\frac{f_1(t)}{f(t)}dt\}⩽1& \text{if }{M}^{\prime }<0\end{array}$$

(2.4)

for all $x\in [a,b]$. Since $f\in C[a,b]$ and $|f|>0$, there exist constants $0<m⩽M$ such that $m⩽|f(x)|⩽M$ for all $x\in [a,b]$. Hence, (2.4) implies that

$$\frac{1}{M}{e}^{|m^{\prime }|(a-b)}⩽|F(x)|⩽\frac{1}{m}{e}^{|M^{\prime }|(b-a)}$$

for all $x\in [a,b]$. It follows from (2.3) that

$$\begin{array}{rl}|y(x)-u(x)|& ⩽\epsilon (b-a)\frac{1}{|F(x)|}{\int }_x^b\left|\frac{F(t)}{f(t)}\right|dt\\ ⩽\epsilon {(b-a)}^2\frac{M}{m^2}{e}^{(|m^{\prime }|+|M^{\prime }|)(b-a)}\end{array}$$

for all $x\in [a,b]$. □